ខាងក្រោមនេះជាតារាងអាំងតេក្រាល (ព្រីមីទីវ) នៃ អនុគមន៍ត្រីកោណមាត្រ។ សំរាប់អាំងតេក្រាលដែលមានអិចស្ប៉ូណង់ស្យែលនៃអនុគមន៍ត្រីកោណមាត្រ សូមមើលតារាងអាំងតេក្រាលនៃអនុគមន៍អិចស្ប៉ូណង់ស្យែល។ សំរាប់តារាងពេញលេញសូមមើលតារាងអាំងតេក្រាល។. សូមមើលផងដែរ អាំងតេក្រាលត្រីកោណមាត្រ។
នៅក្នុងគ្រប់រូបមន្តខាងក្រោម a ជាចំនួនថេរខុសពីសុន្យ ហើយ C ជា ថេររបស់អាំងតេក្រាល។
![{\displaystyle \int \sin ax\;dx=-{\frac {1}{a}}\cos ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a73eb45066dbbef44abf783f8cac45b6e568f04)
![{\displaystyle \int \sin ^{2}{ax}\;dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/293755ccc64ede76a76570cda68f8bc2eead8f22)
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![{\displaystyle \int {\frac {dx}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/558a7998ef61c2f451a53dc7168710e503e42f8c)
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![{\displaystyle \int x\sin ax\;dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da65650f139648d3bb59885918799df0c6119f7b)
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![{\displaystyle \int {\frac {\sin ax}{x}}dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe560e92679d92b5099e69ce4be2598e3c2f5ea7)
![{\displaystyle \int {\frac {\sin ax}{x^{n}}}dx=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03e7a44d70f97f2ffc9285a1cac5ef28d06f483)
![{\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8439ef42a168ed7e05a7efea83b205790ceb59)
![{\displaystyle \int {\frac {x\;dx}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2bd3d7a463e6f185985d416b060c066837a744)
![{\displaystyle \int {\frac {x\;dx}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b160118dfa4c5a05cba9a2ebc6526cb1064c9eb)
![{\displaystyle \int {\frac {\sin ax\;dx}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e00c6763dad11fcec444a0db9d7fa534dea659fb)
![{\displaystyle \int \cos ax\;dx={\frac {1}{a}}\sin ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23fe2f09db08af7b70bff8e1f52bbc26d4e5e48b)
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![{\displaystyle \int x\cos ax\;dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a224cadde3a4fef31043b5aedd948c765db94c2)
![{\displaystyle \int \cos ^{2}{ax}\;dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e16ba54bc0559f3279fb72e7215f23e4edde65)
![{\displaystyle \int x^{n}\cos ax\;dx={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d842c2fcaf2158d857e9b67d5da59fca2135497b)
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![{\displaystyle \int {\frac {\cos ax}{x}}dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17e94201a5023bd1897254a8bf7fc4606958637c)
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![{\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5013bc2428b1006b40c999d6b427a36f5cf0620)
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![{\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/110e912c95a902109556eaf7954dcde0571fc4c8)
![{\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea932362481541732a94cacdd2885997cdbe7598)
![{\displaystyle \int {\frac {x\;dx}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c75e60562bf09238b854d35ab35bdcbf5c8510)
![{\displaystyle \int {\frac {x\;dx}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5563ed4ebe078ad8616ab89c39b0f832067bdc)
![{\displaystyle \int {\frac {\cos ax\;dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58b792752178d49805802fc55c5a32ff79fc2da0)
![{\displaystyle \int {\frac {\cos ax\;dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1feca0aca8542bf452ecdba3bf778ef0dad786)
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![{\displaystyle \int \tan ax\;dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec533469c6f4008af58416899e6afccbfed393f)
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![{\displaystyle \int {\frac {dx}{\tan ax}}={\frac {1}{a}}\ln |\sin ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/865a88c71d447aa57caa031b819e24f3f76a1f38)
![{\displaystyle \int {\frac {dx}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1711694feda807b6b6c4d19c1ccda303ef0c02b)
![{\displaystyle \int {\frac {dx}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acaa4b1fa8ad7c5675d289c229146ea22ebb1ad)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/401dd109ce2332e369ea8b45ad3e3d8165435b6e)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3c42a4efe9309b306895d993b1980ddda5e5c)
![{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0eae695334d259040d728b565ca374a2c89380)
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![{\displaystyle \int \sec ^{n}{x}\,dx={\frac {\sec ^{n-2}{x}\tan {x}}{n-1}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c22e8160f1fca39c014c254d13a5420a2b5ce8e)
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acabbd90de19b0d361d572dce3398a57c9d653f)
![{\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d51921c472f398df353d0db33a9af2ebe5fbf3fb)
![{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417803af6cef8535c9b9ee74f75a20ab4180fac0)
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![{\displaystyle \int \cot ax\;dx={\frac {1}{a}}\ln |\sin ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a390207aa90027cf4e88003ad404e5be9c3ce9)
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![{\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb992449b8d41c9ad694ebc79755b1bc6f05b721)
![{\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae83987f4461f7cefa25e65f6583476d6f6aa1a7)
![{\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99f9f4158d86f68a6f22ac0b494b8df2a009d24)
![{\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e0e3cebd7eac046797eefb5e8be824a6ec6008)
![{\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{n-1}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4a31631ace2155c341f3a42e944454f4d2525b)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20dbc04e3f7127782e7b005abd8ee54505f6a3c3)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16fc2c648278180416aab94d34a800a261298de)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bed95877232b1041ff194e4431e3085f8d94bb)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5878e7de15317f1a758a684cd716299d7704f5)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ceadea9e2db1da77cc7cc229f5dc509b1367de1)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f23dfa71a079b3aeeab5209cff8bf9b3cfbb33d)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b85fee1b43ee4e960660d5506f2fbabee8b8b51f)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28889f39f656ab104a2636f836512a46015bc42a)
![{\displaystyle \int \sin ax\cos ax\;dx={\frac {1}{2a}}\sin ^{2}ax+c\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91982650837db8088a6e85410cc1445732288a76)
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- ផងដែរ:
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![{\displaystyle \int {\frac {dx}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf404f3f55cfd283adc68644b179ac6bab6f24d7)
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![{\displaystyle \int {\frac {\sin ^{2}ax\;dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/585f6083fc3cb9c10c4ecd369500ce5902de34c6)
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- ផងដែរ:
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- ផងដែរ:
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![{\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbc213a769efe305eaa42857879e989b250e33f)
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![{\displaystyle \int \sin ax\tan ax\;dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8db0ad9f8d3768112f104d143d00d48a4a10a30)
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អាំងតេក្រាលដែលមានគោលឆ្លុះគ្នា[កែប្រែ]
![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33469f374c9c3af903d9607671dcfdf18c1a5077)
![{\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx=2\sin {c}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43f875b81a6b5c38d2d0a7f54fbe9ae958b47526)
![{\displaystyle \int _{-c}^{c}\tan {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a562f35f07612ecdd0e562ba5e93b728320032)